The Fundamental Theorem of Calculus
微积分calculus英文单词
微积分英语单词Absolute convergence ：绝对收敛Absolute extreme values ：绝对极值Absolute maximum and minimum ：绝对极大与极小Absolute value ：绝对值Absolute value function ：绝对值函数Acceleration ：加速度Antiderivative ：反导数Approximate integration ：近似积分Approximation ：逼近法Arc length ：弧长Area ：面积Asymptote ：渐近线Average speed ：平均速率Average velocity ：平均速度Axes, coordinate ：坐标轴Axes of ellipse ：椭圆之轴at a point ：在一点处之连续性as the slope of a tangent ：导数看成切线之斜率by differentials ：用微分逼近between curves ：曲线间之面积Binomial series ：二项级数Cartesian coordinates ：笛卡儿坐标一般指直角坐标Cartesian coordinates system ：笛卡儿坐标系Cauch’s Mean Value Theorem ：柯西均值定理Chain Rule ：连锁律Change of variables ：变数变换Circle ：圆Circular cylinder ：圆柱Closed interval ：封闭区间Coefficient ：系数Composition of function ：函数之合成Compound interest ：复利Concavity ：凹性Conchoid ：蚌线Cone ：圆锥Constant function ：常数函数Constant of integration ：积分常数Continuity ：连续性Continuous function ：连续函数Convergence ：收敛Coordinate ：s ：坐标Cartesian ：笛卡儿坐标cylindrical ：柱面坐标Coordinate axes ：坐标轴Coordinate planes ：坐标平面Cosine function ：余弦函数Critical point ：临界点Cubic function ：三次函数Curve ：曲线Cylinder ：圆柱Cylindrical Coordinates ：圆柱坐标Distance ：距离Divergence ：发散Domain ：定义域Dot product ：点积Double integral ：二重积分Decreasing function ：递减函数Decreasing sequence ：递减数列Definite integral ：定积分Degree of a polynomial ：多项式之次数Density ：密度Derivative ：导数Determinant ：行列式Differentiable function ：可导函数Differential ：微分Differential equation ：微分方程Differentiation ：求导法Directional derivatives ：方向导数Discontinuity ：不连续性Disk method ：圆盘法domain of ：导数之定义域differential ：微分学Ellipse ：椭圆Ellipsoid ：椭圆体Epicycloid ：外摆线Equation ：方程式Even function ：偶函数Expected Valued ：期望值Exponential Function ：指数函数Exponents , laws of ：指数率Extreme value ：极值Extreme Value Theorem ：极值定理Factorial ：阶乘First Derivative Test ：一阶导数试验法First octant ：第一卦限Focus ：焦点Fractions ：分式Function ：函数Fundamental Theorem of Calculus ：微积分基本定理from the left ：左连续from the right ：右连续Geometric series ：几何级数Gradient ：梯度Graph ：图形Green Formula ：格林公式Half-angle formulas ：半角公式Harmonic series ：调和级数Helix ：螺旋线Higher Derivative ：高阶导数Horizontal asymptote ：水平渐近线Horizontal line ：水平线Hyperbola ：双曲线Hyperboloid ：双曲面horizontal ：水平渐近线Implicit differentiation ：隐求导法Implicit function ：隐函数Improper integral ：瑕积分Increasing/Decreasing Test ：递增或递减试验法Increment ：增量Increasing Function ：增函数Indefinite integral ：不定积分Independent variable ：自变数Indeterminate from ：不定型Inequality ：不等式Infinite point ：无穷极限Infinite series ：无穷级数Inflection point ：反曲点Instantaneous velocity ：瞬时速度Integer ：整数Integral ：积分Integrand ：被积分式Integration ：积分Integration by part ：分部积分法Intercepts ：截距Intermediate value of Theorem ：中间值定理Interval ：区间Inverse function ：反函数Inverse trigonometric function ：反三角函数Iterated integral ：逐次积分integral ：积分学implicit ：隐求导法Laplace transform ：Leplace 变换Law of Cosines ：余弦定理Least upper bound ：最小上界Left-hand derivative ：左导数Left-hand limit ：左极限Lemniscate ：双钮线Length ：长度Level curve ：等高线L'Hospital's rule ：洛必达法则Limacon ：蚶线Limit ：极限Linear approximation：线性近似Linear equation ：线性方程式Linear function ：线性函数Linearity ：线性Linearization ：线性化Line in the plane ：平面上之直线Line in space ：空间之直线Lobachevski geometry ：罗巴切夫斯基几何Local extremum ：局部极值Local maximum and minimum ：局部极大值与极小值Logarithm ：对数Logarithmic function ：对数函数linear ：线性逼近法Maximum and minimum values ：极大与极小值Mean Value Theorem ：均值定理Multiple integrals ：重积分Multiplier ：乘子Natural exponential function ：自然指数函数Natural logarithm function ：自然对数函数Natural number ：自然数Normal line ：法线Normal vector ：法向量Number ：数of a function ：函数之连续性on an interval ：在区间之连续性Octant ：卦限Odd function ：奇函数One-sided limit ：单边极限Open interval ：开区间Optimization problems ：最佳化问题Order ：阶Ordinary differential equation ：常微分方程 Origin ：原点Orthogonal ：正交的Parabola ：拋物线Parabolic cylinder ：抛物柱面Paraboloid ：抛物面Parallelepiped ：平行六面体Parallel lines ：并行线Parameter ：参数Partial derivative ：偏导数Partial differential equation ：偏微分方程 Partial fractions ：部分分式Partial integration ：部分积分Partiton ：分割Period ：周期Periodic function ：周期函数Perpendicular lines ：垂直线Piecewise defined function ：分段定义函数 Plane ：平面Point of inflection ：反曲点Polar axis ：极轴Polar coordinate ：极坐标Polar equation ：极方程式Pole ：极点Polynomial ：多项式Positive angle ：正角Point-slope form ：点斜式Power function ：幂函数Product ：积polar ：极坐标partial ：偏导数partial ：偏微分方程partial ：偏微分法Quadrant ：象限Quotient Law of limit ：极限的商定律Quotient Rule ：商定律rectangular ：直角坐标Radius of convergence ：收敛半径Range of a function ：函数的值域Rate of change ：变化率Rational function ：有理函数Rationalizing substitution ：有理代换法Rationalizing substitution ：有理代换法Rational number ：有理数Real number ：实数Rectangular coordinates ：直角坐标Rectangular coordinate system ：直角坐标系Relative maximum and minimum ：相对极大值与极小值Revenue function ：收入函数Revolution, solid of ：旋转体Revolution, surface of ：旋转曲面Riemann Sum ：黎曼和Riemannian geometry ：黎曼几何Right-hand derivative ：右导数Right-hand limit ：右极限Root ：根Saddle point ：鞍点Scalar ：纯量Secant line ：割线Second derivative ：二阶导数Second Derivative Test ：二阶导数试验法Second partial derivative ：二阶偏导数Sector ：扇形Sequence ：数列Series ：级数Set ：集合Shell method ：剥壳法Sine function ：正弦函数Singularity ：奇点Slant asymptote ：斜渐近线Slope ：斜率Slope-intercept equation of a line ：直线的斜截式Smooth curve ：平滑曲线Smooth surface ：平滑曲面Solid of revolution ：旋转体Space ：空间Speed ：速率Spherical coordinates ：球面坐标Squeeze Theorem ：夹挤定理Step function ：阶梯函数Strictly decreasing ：严格递减Strictly increasing ：严格递增Sum ：和Surface ：曲面Surface integral ：面积分Surface of revolution ：旋转曲面Symmetry ：对称slant ：斜渐近线spherical ：球面坐标Tangent function ：正切函数Tangent line ：切线Tangent plane ：切平面Tangent vector ：切向量Total differential ：全微分Trigonometric function ：三角函数Trigonometric integrals ：三角积分Trigonometric substitutions ：三角代换法Tripe integrals ：三重积分term by term ：逐项求导法under a curve ：曲线下方之面积vertical ：垂直渐近线Value of function ：函数值Variable ：变数Vector ：向量Velocity ：速度Vertical asymptote ：垂直渐近线Volume ：体积X-axis ：x 轴x-coordinate ：x 坐标x-intercept ：x 截距Zero vector ：函数的零点Zeros of a polynomial ：多项式的零点。
实数英语知识点总结
实数英语知识点总结In mathematics, real numbers are a fundamental concept that encompasses all rational and irrational numbers. Real numbers include integers, fractions, decimals, and irrational numbers such as √2 and π. The real number system is a crucial concept in algebra, calculus, and other branches of mathematics, and it forms the foundation for many mathematical principles and theories.Properties of Real NumbersReal numbers have several key properties that distinguish them from other types of numbers. These properties include:1. Closure: The sum, difference, product, and quotient of any two real numbers are also real numbers. In other words, the real number system is closed under addition, subtraction, multiplication, and division.2. Commutative property: The order in which real numbers are added or multiplied does not affect the result. For example, a + b = b + a and a * b = b * a for any real numbers a and b.3. Associative property: The grouping of real numbers in addition and multiplication does not affect the result. For example, (a + b) + c = a + (b + c) and (a * b) * c = a * (b * c) for any real numbers a, b, and c.4. Distributive property: Multiplication distributes over addition. In other words, a * (b + c) = a * b + a * c for any real numbers a, b, and c.5. Identity elements: The real number 0 serves as the additive identity, meaning that a + 0 =a for any real number a. The real number 1 serves as the multiplicative identity, meaning that a * 1 = a for any real number a.6. Inverses: Every real number a has an additive inverse -a such that a + (-a) = 0, and every nonzero real number a has a multiplicative inverse 1/a such that a * (1/a) = 1.These properties make the real number system a powerful and versatile tool for solving mathematical problems and modeling real-world phenomena.Types of Real NumbersReal numbers can be categorized into several types based on their properties, including:1. Rational numbers: Rational numbers are real numbers that can be expressed as a fraction of two integers, where the denominator is not zero. Examples of rational numbers include 1/2, -3/4, and 5.2. Irrational numbers: Irrational numbers are real numbers that cannot be expressed as a fraction of two integers. Instead, they have an infinite non-repeating decimal expansion. Examples of irrational numbers include √2, π, and e.3. Integers: Integers are real numbers that include all positive and negative whole numbers, as well as zero. Examples of integers include -3, 0, and 7.4. Whole numbers: Whole numbers are real numbers that include all positive whole numbers and zero. Examples of whole numbers include 0, 1, and 7.5. Natural numbers: Natural numbers are real numbers that include all positive whole numbers. Examples of natural numbers include 1, 2, and 3.The real number system encompasses all these types of numbers and provides a framework for understanding their relationships and properties.Operations on Real NumbersReal numbers can be operated on using various mathematical operations, including addition, subtraction, multiplication, division, and exponentiation. These operations follow specific rules and properties that govern how real numbers interact with each other.Addition and Subtraction: When adding or subtracting real numbers, the numbers are combined based on their sign. If the numbers have the same sign, their absolute values are added or subtracted, and the result takes on that sign. If the numbers have different signs, their absolute values are subtracted, and the result takes on the sign of the number with the larger absolute value.Multiplication and Division: When multiplying or dividing real numbers, the numbers retain their signs, and the product or quotient takes on the sign of the numbers being operated on. In division, the denominator cannot be zero, as division by zero is undefined in the real number system.Exponentiation: Exponentiation involves raising a real number to a power, or exponent. For example, a^b represents a raised to the power of b. Exponentiation follows specific rules, such as the product rule (a^b * a^c = a^(b+c)), the quotient rule (a^b / a^c = a^(b-c)), and the power rule ((a^b)^c = a^(b*c)).These operations and their corresponding rules provide a systematic way to manipulate and calculate with real numbers, leading to the development of algebraic expressions, equations, and functions.Real Numbers and Number LinesReal numbers can be visualized and represented using a number line, which is a horizontal line that extends infinitely in both directions. On the number line, each real number is assigned a point based on its value, with positive numbers to the right of zero and negative numbers to the left of zero.The number line provides a geometric representation of the order and magnitude of real numbers, making it easier to compare and order real numbers, perform arithmetic operations, and solve inequalities.For example, on a number line, the real number 3 is located to the right of 2, indicating that 3 is greater than 2. Similarly, the real number -5 is located to the left of -3, indicating that -5 is less than -3.In addition to representing individual real numbers, number lines can be used to depict intervals, or sets of real numbers between two given values. For example, the interval [a, b] includes all real numbers x such that a ≤ x ≤ b, while the interval (a, b) includes all real numbers x such that a < x < b.The use of number lines provides a visual and intuitive way to understand and work with real numbers, enhancing the conceptual understanding and application of mathematical concepts.Real Numbers in CalculusReal numbers play a central role in calculus, a branch of mathematics that deals with the study of change and motion. In calculus, real numbers are used to represent quantities such as distance, time, velocity, and acceleration, allowing for the formulation and solution of mathematical models for physical phenomena.The concepts of limits, continuity, derivatives, and integrals in calculus are all based on real numbers and their properties. For example, the limit of a function as its input approaches a real number represents the behavior of the function near that number, while the derivative of a function at a real number represents the rate of change of the function at that number.In addition, real numbers are used to define and analyze mathematical functions, which are essential for describing relationships between variables and making predictions about real-world phenomena. Functions such as polynomials, exponential functions, logarithmic functions, and trigonometric functions all operate on real numbers and form the basis for mathematical modeling and analysis in calculus.Furthermore, the fundamental theorem of calculus, which relates the concepts of differentiation and integration, is formulated and proven using real numbers. This theorem provides a powerful tool for computing areas, volumes, and other quantities based on their rates of change, and it has widespread applications in science, engineering, and economics.Overall, real numbers are indispensable in the study and application of calculus, providing a solid mathematical foundation for understanding and analyzing the behavior of natural and man-made systems.ConclusionReal numbers are a fundamental and versatile concept in mathematics, encompassing rational and irrational numbers, integers, fractions, and decimals. The properties of real numbers, including closure, commutativity, associativity, and identity elements, make them a powerful tool for solving mathematical problems and modeling real-world phenomena. Through operations such as addition, subtraction, multiplication, division, and exponentiation, real numbers can be manipulated and calculated to derive new relationships and insights. The use of number lines provides a visual representation of real numbers and their relationships, enhancing the conceptual understanding and application of mathematical concepts.In disciplines such as calculus, real numbers serve as the basis for understanding and analyzing change and motion, providing a framework for the study of limits, derivatives, integrals, and functions.Overall, real numbers form an essential foundation for mathematics and its applications, shaping the way we understand and interact with the world around us.。
定积分的英文专用名词
定积分的英文专用名词Definite Integral: A Deep Dive into Its Concepts and Applications.In the realm of mathematics, the definite integral occupies a pivotal position, bridging the gap between discrete and continuous functions. It represents a fundamental tool for understanding the properties of functions and for solving a wide range of practical problems. This article aims to delve into the essence of the definite integral, exploring its definitions, properties, applications, and the underlying theories that govern it.1. Introduction to Definite Integrals.The definite integral is a mathematical operation that assigns a real number to a given function on a closed interval. It is denoted as ∫[a,b]f(x)dx, where [a,b] is the closed interval and f(x) is the function beingintegrated. This notation represents the accumulation of infinitesimal changes in the function over the specified interval.The definite integral is closely related to the indefinite integral or the antiderivative. While the indefinite integral represents a family of functions that differ by a constant, the definite integral provides a specific numerical value for a given function over a specific interval.2. Properties of Definite Integrals.The definite integral possesses several important properties that govern its behavior and applications. Some of the key properties include:Linearity: The definite integral distributes over addition and scalar multiplication, allowing us to integrate complex expressions by breaking them down into simpler parts.Interval Additivity: If an interval [a,b] is divided into two subintervals [a,c] and [c,b], then the integral over the entire interval is equal to the sum of the integrals over the subintervals.Constant Factor Rule: If a constant k is multiplied with a function f(x) within the integral, then the integral of kf(x) is equal to k times the integral of f(x).Comparison Theorem: If f(x) ≤ g(x) for all x in the interval [a,b], then the integral of f(x) over [a,b] is less than or equal to the integral of g(x) over the same interval.These properties provide a solid foundation for solving complex integration problems and for understanding the behavior of functions under integration.3. Applications of Definite Integrals.The definite integral finds applications in various fields of mathematics, science, and engineering. Some ofits key applications include:Area Calculation: The most fundamental application of the definite integral is in calculating the area under a curve between two given points. This is achieved by integrating the function representing the curve over the corresponding interval.Volume Calculation: The definite integral can be extended to calculate volumes of three-dimensional objects, such as solids of revolution and regions between surfaces.Length of Curves: The integral can be used to compute the length of a curve in the plane or in space, by integrating the square root of the sum of squares of the derivatives of the curve's coordinates.Physics and Engineering: The definite integral plays a crucial role in physics and engineering, particularly in areas such as mechanics, electromagnetism, and fluid dynamics. It is used to calculate quantities like displacement, velocity, acceleration, force, work, energy,and momentum.Probability and Statistics: In probability theory and statistics, the definite integral is employed to calculate probabilities, expected values, and other statistical measures.4. Theories Underlying Definite Integrals.The theory of definite integrals is rooted in the fundamental theorem of calculus, which establishes a link between differentiation and integration. This theorem states that if F(x) is an antiderivative of f(x), then the integral of f(x) from a to b is equal to F(b) F(a). This theorem provides a practical method for evaluating definite integrals by finding antiderivatives of the integrated functions.In addition to the fundamental theorem, other important theories include the convergence theorems for infinite series and improper integrals, which extend the concept of definite integrals to infinite intervals and infinite sums.These theories provide a rigorous foundation for understanding the behavior of integrals and for solving complex integration problems.5. Conclusion.The definite integral is a powerful tool that lies at the heart of mathematics and its applications. Its ability to measure areas, volumes, lengths, and other quantities has made it indispensable in various fields ranging from physics and engineering to economics and finance. By delving into its definitions, properties, applications, and underlying theories, we can gain a deeper understanding of this fundamental concept and appreciate its value in addressing real-world problems.。
高三英语微积分基础单选题20题及答案
高三英语微积分基础单选题20题及答案1. In calculus, the derivative of a constant is _____.A. zeroB. oneC. itselfD. undefined答案：A。
常数的导数是零。
选项B“one”错误，常数的导数不是1。
选项C“itself”错误，常数的导数不是它本身。
选项D“undefined”错误，常数的导数是确定的，为零。
2. The process of finding the derivative is called _____.A. integrationB. differentiationC. summationD. multiplication答案：B。
求导数的过程叫做微分。
选项A“integration”是积分。
选项C“summation”是求和。
选项D“multiplication”是乘法。
3. If y = x², then the derivative of y with respect to x is _____.A. 2xB. x²C. 2x²D. x/2答案：A。
y = x²的导数是2x。
选项B“x²”错误，不是它本身。
选项C“2x²”错误，系数错误。
选项D“x/2”错误，计算错误。
4. The integral of a constant times a function is equal to the constant times the integral of the function. This is known as _____.A. the power ruleB. the product ruleC. the chain ruleD. the constant multiple rule答案：D。
常数乘以函数的积分等于常数乘以函数的积分，这被称为常数倍数法则。
算子总结;哈密尔顿算子;拉普拉斯算子
算子总结;哈密尔顿算子;拉普拉斯算子-标准化文件发布号：（9556-EUATWK-MWUB-WUNN-INNUL-DDQTY-KII∇：向量微分算子、哈密尔顿算子、Nabla算子、劈形算子，倒三角算子是一个微分算子。
Strictly speaking, ∇del is not a specific operator, but rather a convenient mathematical notation for those three operators, that makes many equations easier to write and remember. The del symbol can be interpreted as a vector of partial derivative operators, and its three possible meanings—gradient, divergence, and curl—can be formally viewed as the product of scalars, dot product, and cross product, respectively, of the del "operator" with the field.Δ、∇2 or ∇·∇：拉普拉斯算子（Laplace operator），定义为梯度（▽f）的散度（▽·f）。
,grad F=▽F，梯度（gradient），标量场的梯度是一个向量场。
标量场中某一点上的梯度指向标量场增长最快的方向，梯度的长度是这个最大的变化率。
▽f=div F=▽·F,散度(divergence),是算子▽点乘向量函数,矢量场的散度是一个标量函数，与求梯度正好相反,div F表示在点M处的单位体积内散发出来的矢量F的通量，描述了通量源的密度，可用表征空间各点矢量场发散的强弱程度。
AP Calculus BC AP微积分BC考试内容授课大纲
AP Calculus BC AP微积分BC考试内容授课大纲AP微积分BC包含AB的内容，难度也远高于AB，大多数学校也更愿意接受BC的成绩。
一般需要花费1年的时间来完成。
微积分是除了部分文科和艺术类专业外，绝大多数专业都要学的基础课程，专业适用面很广。
AP微积分BC考试将测试你对课程单元所涵盖的数学概念的理解，以及你解决问题所用的适当公式和程序的能力，以及用正确的符号交流工作的能力。
部分考试允许使用绘图计算器。
注意：你不能在同一年内同时参加AP微积分AB和微积分BC考试。
考试时长：3hrs 15mins考试时间：Tue, May 4, 2021, 8 AM Local考试分布：Section 1:Multiple Choice多选题，45道题，时长1hr45mins，占比50%Section 2:Free Response自由回答，6道题，时常1hr30mins，占比50%在此之前学生最好掌握以下的知识点：代数、几何、三角学、解析几何、初等函数的课程，特别是理解线性、多项式、有理函数、指数函数、对数函数、三角函数、反三角函数、分段函数，以及数列、级数、极方程。
你应该知道如何画这些函数的图和解。
你还应该熟悉一般函数的代数变换、组合、综合和逆运算。
课程安排：1、Unit 1: 极限和连续性You’ll start to explore how limits will allow you to solve problems involving change学习用极限解决变化问题，并更好地理解函数的数学推理。
Topics may include:•How limits help us to handle change at an instant 极限如何帮助我们处理瞬间的变化•Definition and properties of limits in variousrepresentations各种表示法中极限的定义和性质•Definitions of continuity of a function at apoint and over a domain函数在一点上和区间上连续性的定义•Asymptotes and limits at infinity渐近线和无穷极限•Reasoning using the Squeeze theorem and theIntermediate Value Theorem用夹逼定理和介值定理进行推理考试占比4%–7%Unit 2:微分：定义与基本性质Unit 2: Differentiation: Definition and Fundamental PropertiesTopics may include:•Defining the derivative of a function at a point and as a function定义在一点处函数的导数和导函数•Connecting differentiability and continuity可微性与连续性•Determining derivatives for elementary functions微分运算法则•Applying differentiation rules微分运算法则考试占比4%–7%Unit 3: Differentiation: Composite, Implicit, and Inverse Functions微分：复合函数、隐函数和反函数掌握使用链式法则，学习新的微分技巧和高阶导数Topics may include:· The chain rule for differentiating posite functions 微分复合函数的链式法则· Implicit differentiation隐函数微分· Differentiation of general and particular inverse functions区分一般反函数和特殊反函数· Determining higher-order derivatives of functions 函数的高阶导数求法考试占比4%–7%Unit 4: Contextual Applications of Differentiation微分的实境应用利用导数设置和解决瞬时变化率的实际问题，利用数学推理求不定式的极限。
微积分基本定理—牛顿莱布尼茨公式
而原函数是与导函数互逆的一个概念，本质上属于
微分学，形式上看，与定积分没有关系。 Newton 和 Leibniz 却发现了这两个概念之间的内在联系：
函数在一个区间上的定积分等于它的原函数在该区间上的增量。 从此微分学与积分学形成一门完整学科——微积分学。
（2）为 定积分的计算提供了一个有效方法. 如果被积函数连续且其原函数易于求得，则只需 先求出原函数，再将上限和下限代入原函数后相减：
定理2 如果函数 f (x)在[a,b]上连续, 函数 F ( x)是 f ( x)
的一个原函数，则
∫b f ( x) dx = F (b) − F (a). a
（上式称为牛顿—莱布尼茨公式，也叫微积分基本公式）
证 因F ( x)与 Φ ( x) = ∫ x f (t )dt 都是 f ( x) 的原函数， a
证 设 F (t ) 是 f (t ) 的原函数，由 N-L 公式，得
∫ϕ(x)
ψ (x)
f
(t ) dt
=
[
F
(t
)]ψϕ
(x) ( x)
=
F
ϕ
( x)
−
F
ψ
( x)
，
于是，
∫ ϕ(x)
ψ ( x)
f
(t)
dt
′
=
F′ ϕ
( x)ϕ′(
x)
−
F′
ψ
(
x)ψ
′(x)
= f ϕ ( x)ϕ′( x) − f ψ ( x)ψ ′( x).
y
y = f (t)
定义了以 x 为自变量的一个
函数，记为Φ ( x), 即
Φ(x)
高等数学英文版课件PPT 05 Integrals
Example 3 Find the area under the cosine curve from 0 to b,
where 0 b / 2.
Solution We choose a regular partition P so that
||P||=b/n
and we choose xi to be the right-hand endpoint of the ith sub-
Chapter 5
Integrals
5.2 Area 5.3 The Definite Integral 5.4 The Fundamental Theorem of Calculus 5.5 The Substitution Rule
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4.2 Area
Area Problem: Find the area of the region S that lies under the curve y=f(x) from a to b.(see Figure 1)
n
n
Ai f (xi)xi
i 1
i 1
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Figure 3
y y=f(x)
S1 S2
Si
Sn
oa
Xi-1
Xi
x b
approximated by
y y=f(x)
Figure 4
R1 R2
o a x1 x2
Ri
Rn
Xi-1
Xi
xi
xn b
x
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Example 2: Find the area under the parabola y=x2+1 from 0 to 2.
微积分中的关键术语
微积分中的关键术语微积分中的关键术语常量：习惯用字母a,b,c,d等表示；变量：习惯用字母x,y,z,u,v,w等表示.函数关系：变量与变量之间的对应关系；极限：变量的变化趋势；导数：变量变化的快慢程度（变化率问题）；微分：函数在某一点处，当自变量有一个微小的改变量时，函数所取得的相应改变量的大小。
dy；y≈积分学：已知某个函数F（x）的导函数f(x),求F（x），使()()x 'F=xfValue of function ：函数值Variable ：变数Vector ：向量Velocity ：速度Vertical asymptote ：垂直渐近线V olume ：体积X-axis ：x轴x-coordinate ：x坐标x-intercept ：x截距Zero vector ：函数的零点Zeros of a polynomial ：多项式的零点T:Tangent function ：正切函数Tangent line ：切线Tangent plane ：切平面Tangent vector ：切向量Total differential ：全微分Trigonometric function ：三角函数Trigonometric integrals ：三角积分Trigonometric substitutions ：三角代换法Tripe integrals ：三重积分S:Saddle point ：鞍点Scalar ：纯量Secant line ：割线Second derivative ：二阶导数Second Derivative Test ：二阶导数试验法Second partial derivative ：二阶偏导数Sector ：扇形Sequence ：数列Series ：级数Set ：集合Shell method ：剥壳法Sine function ：正弦函数Singularity ：奇点Slant asymptote ：斜渐近线Slope ：斜率Slope-intercept equation of a line ：直线的斜截式Smooth curve ：平滑曲线Smooth surface ：平滑曲面Solid of revolution ：旋转体Space ：空间Speed ：速率Spherical coordinates ：球面坐标Squeeze Theorem ：夹挤定理Step function ：阶梯函数Strictly decreasing ：严格递减Strictly increasing ：严格递增Sum ：和Surface ：曲面Surface integral ：面积分Surface of revolution ：旋转曲面Symmetry ：对称R:Radius of convergence ：收敛半径Range of a function ：函数的值域Rate of change ：变化率Rational function ：有理函数Rationalizing substitution ：有理代换法Rational number ：有理数Real number ：实数Rectangular coordinates ：直角坐标Rectangular coordinate system ：直角坐标系Relative maximum and minimum ：相对极大值与极小值Revenue function ：收入函数Revolution , solid of ：旋转体Revolution , surface of ：旋转曲面Riemann Sum ：黎曼和Riemannian geometry ：黎曼几何Right-hand derivative ：右导数Right-hand limit ：右极限Root ：根P、Q:Parabola ：拋物线Parabolic cylinder ：抛物柱面Paraboloid ：抛物面Parallelepiped ：平行六面体Parallel lines ：并行线Parameter ：参数Partial derivative ：偏导数Partial differential equation ：偏微分方程Partial fractions ：部分分式Partial integration ：部分积分Partiton ：分割Period ：周期Periodic function ：周期函数Perpendicular lines ：垂直线Piecewise defined function ：分段定义函数Plane ：平面Point of inflection ：反曲点Polar axis ：极轴Polar coordinate ：极坐标Polar equation ：极方程式Pole ：极点Polynomial ：多项式Positive angle ：正角Point-slope form ：点斜式Power function ：幂函数Product ：积Quadrant ：象限Quotient Law of limit ：极限的商定律Quotient Rule ：商定律M、N、O:Maximum and minimum values ：极大与极小值Mean Value Theorem ：均值定理Multiple integrals ：重积分Multiplier ：乘子Natural exponential function ：自然指数函数Naturallogarithm function ：自然对数函数Natural number ：自然数Normal line ：法线Normal vector ：法向量Number ：数Octant ：卦限Odd function ：奇函数One-sided limit ：单边极限Open interval ：开区间Optimization problems ：最佳化问题Order ：阶Ordinary differential equation ：常微分方程Origin ：原点Orthogonal ：正交的L:Laplace transform ：Leplace 变换Law of Cosines ：余弦定理Least upper bound ：最小上界Left-hand derivative ：左导数Left-hand limit ：左极限Lemniscate ：双钮线Length ：长度Level curve ：等高线L'Hospital's rule ：洛必达法则Limacon ：蚶线Limit ：极限Linear approximation：线性近似Linear equation ：线性方程式Linear function ：线性函数Linearity ：线性Linearization ：线性化Line in the plane ：平面上之直线Line in space ：空间之直线Lobachevski geometry ：罗巴切夫斯基几何Local extremum ：局部极值Local maximum and minimum ：局部极大值与极小值Logarithm ：对数Logarithmic function ：对数函数I:Implicit differentiation ：隐求导法Implicit function ：隐函数Improper integral ：瑕积分Increasing/Decreasing Test ：递增或递减试验法Increment ：增量Increasing Function ：增函数Indefinite integral ：不定积分Independent variable ：自变数Indeterminate from ：不定型Inequality ：不等式Infinite point ：无穷极限Infinite series ：无穷级数Inflection point ：反曲点Instantaneous velocity ：瞬时速度Integer ：整数Integral ：积分Integrand ：被积分式Integration ：积分Integration by part ：分部积分法Intercepts ：截距Intermediate value of Theorem ：中间值定理Interval ：区间Inverse function ：反函数Inverse trigonometric function ：反三角函数Iterated integral ：逐次积分H:Higher mathematics 高等数学/高数E、F、G、H:Ellipse ：椭圆Ellipsoid ：椭圆体Epicycloid ：外摆线Equation ：方程式Even function ：偶函数Expected Valued ：期望值Exponential Function ：指数函数Exponents , laws of ：指数率Extreme value ：极值Extreme Value Theorem ：极值定理Factorial ：阶乘First Derivative Test ：一阶导数试验法First octant ：第一卦限Focus ：焦点Fractions ：分式Function ：函数Fundamental Theorem of Calculus ：微积分基本定理Geometric series ：几何级数Gradient ：梯度Graph ：图形Green Formula ：格林公式Half-angle formulas ：半角公式Harmonic series ：调和级数Helix ：螺旋线Higher Derivative ：高阶导数Horizontal asymptote ：水平渐近线Horizontal line ：水平线Hyperbola ：双曲线Hyper boloid ：双曲面D:Decreasing function ：递减函数Decreasing sequence ：递减数列Definite integral ：定积分Degree of a polynomial ：多项式之次数Density ：密度Derivative ：导数of a composite function ：复合函数之导数of a constant function ：常数函数之导数directional ：方向导数domain of ：导数之定义域of exponential function ：指数函数之导数higher ：高阶导数partial ：偏导数of a power function ：幂函数之导数of a power series ：羃级数之导数of a product ：积之导数of a quotient ：商之导数as a rate of change ：导数当作变率right-hand ：右导数second ：二阶导数as the slope of a tangent ：导数看成切线之斜率Determinant ：行列式Differentiable function ：可导函数Differential ：微分Differential equation ：微分方程partial ：偏微分方程Differentiation ：求导法implicit ：隐求导法partial ：偏微分法term by term ：逐项求导法Directional derivatives ：方向导数Discontinuity ：不连续性Disk method ：圆盘法Distance ：距离Divergence ：发散Domain ：定义域Dot product ：点积Double integral ：二重积分change of variable in ：二重积分之变数变换in polar coordinates ：极坐标二重积分C:Calculus ：微积分differential ：微分学integral ：积分学Cartesian coordinates ：笛卡儿坐标一般指直角坐标Cartesian coordinates system ：笛卡儿坐标系Cauch’s Mean Value Theorem ：柯西均值定理Chain Rule ：连锁律Change of variables ：变数变换Circle ：圆Circular cylinder ：圆柱Closed interval ：封闭区间Coefficient ：系数Composition of function ：函数之合成Compound interest ：复利Concavity ：凹性Conchoid ：蚌线Cone ：圆锥Constant function ：常数函数Constant of integration ：积分常数Continuity ：连续性at a point ：在一点处之连续性of a function ：函数之连续性on an interval ：在区间之连续性from the left ：左连续from the right ：右连续Continuous function ：连续函数Convergence ：收敛interval of ：收敛区间radius of ：收敛半径Convergent sequence ：收敛数列 series ：收敛级数Coordinate：s：坐标Cartesian ：笛卡儿坐标cylindrical ：柱面坐标polar ：极坐标rectangular ：直角坐标spherical ：球面坐标Coordinate axes ：坐标轴Coordinate planes ：坐标平面Cosine function ：余弦函数Critical point ：临界点Cubic function ：三次函数Curve ：曲线Cylinder：圆柱Cylindrical Coordinates ：圆柱坐标A、B:Absolute convergence ：绝对收敛Absolute extreme values ：绝对极值Absolute maximum and minimum ：绝对极大与极小Absolute value ：绝对值Absolute value function ：绝对值函数Acceleration ：加速度Antiderivative ：反导数Approximate integration ：近似积分Approximation ：逼近法by differentials ：用微分逼近linear ：线性逼近法by Simpson’s Rule ：Simpson法则逼近法by the Trapezoidal Rule ：梯形法则逼近法Arbitrary constant ：任意常数Arc length ：弧长Area ：面积under a curve ：曲线下方之面积between curves ：曲线间之面积in polar coordinates ：极坐标表示之面积of a sector of a circle ：扇形之面积of a surface of a revolution ：旋转曲面之面积Asymptote ：渐近线horizontal ：水平渐近线slant ：斜渐近线vertical ：垂直渐近线Average speed ：平均速率Average velocity ：平均速度Axes, coordinate ：坐标轴Axes of ellipse ：椭圆之轴Binomial series ：二项级数。
高等数学（微积分学）专业术语名词概念定理等英汉对照
高等数学（微积分学）专业术语名词概念定理等英汉对照目录第一部分英汉微积分词汇Part 1 English-Chinese Calculus Vocabulary第一章函数与极限Chapter 1 function and Limi t (1)第二章导数与微分Chapter 2 Derivative and Differential (2)第三章微分中值定理Chapter 3 Mean Value theorem of differentials and the Application of Derivatives (3)第四章不定积分Chapter 4 Indefinite Intergrals (3)第五章定积分Chapter 5 Definite Integral (3)第六章定积分的应用Chapter 6 Application of the Definite Integrals (4)第七章空间解析几何与向量代数Chapter 7 Space Analytic Geomertry and Vector Algebra (4) 第八章多元函数微分法及其应用Chapter 8 Differentiation of functions Several variables and Its Application (5)第九章重积分Multiple Integrals (6)第十章曲线积分与曲面积分Chapter 10 Line(Curve ) Integrals and Surface Integral s (6) 第十一章无穷级数Chapter 11 Infinite Series (6)第十二章微分方程Chapter 12 Differential Equation (7)第二部分定理定义公式的英文表达Part 2 English Expression for Theorem,Definition and Formula第一章函数与极限Chapter 1 Function and Limi t (19)1.1映射与函数(Mapping and Function ) (19)1.2数列的极限(Limit of the Sequence of Number) (20)1.3函数的极限(Limit of Function) (21)1.4无穷小与无穷大(Infinitesimal and Inifinity) (23)1.5极限运算法则(Operation Rule of Limit) (24)1.6极限存在准则两个重要的极限(Rule for theExistence of Limits Two Important Limits) (25)1.7无穷小的比较(The Comparison of infinitesimal) (26)1.8函数的连续性与间断点(Continuity of FunctionAnd Discontinuity Points) (28)1.9连续函数的运酸与初等函数的连续性(OperationOf Continuous Functions and Continuity ofElementary Functions) (28)1.10闭区间上联系汗水的性质(Properties ofContinuous Functions on a Closed Interval) (30)第二章导数与数分Chapter2 Derivative and Differential (31)2.1 导数的概念(The Concept of Derivative) (31)2.2 函数的求导法则(Rules for Finding Derivatives) (33)2.3 高阶导数(Higher-order Derivatives) (34)2.4 隐函数及由参数方程所确定的函数的导数相关变化率(Derivatives ofImplicit Functions and Functions Determined by Parametric Equation andCorrelative Change Rate) (34)2.5 函数的微分(Differential of a Function) (35)第三章微分中值定理与导数的应用Chapter 3 Mean Value Theorem of Differentials and theApplication of Derivatives (36)3.1 微分中值定理(The Mean Value Theorem) (36)3.2 洛必达法则(L’Hopital’s Rule) (38)3.3 泰勒公式(Taylor’s Formula) (41)3.4 函数的单调性和曲线的凹凸性(Monotonicityof Functions and Concavity of Curves) (43)3.5 函数的极值与最大最小值(Extrema, Maximaand Minima of Functions) (46)3.6 函数图形的描绘(Graphing Functions) (49)3.7 曲率(Curvature) (50)3.8 方程的近似解(Solving Equation Numerically) (53)第四章不定积分Chapter 4Indefinite Integrals (54)4.1 不定积分的概念与性质(The Concept andProperties of Indefinite Integrals) (54)4.2 换元积分法(Substitution Rule for Indefinite Integrals) (56)4.3 分部积分法(Integration by Parts) (57)4.4 有理函数的积分(Integration of Rational Functions) (58)第五章定积分Chapter 5 Definite Integrals (61)5.1 定积分的概念和性质(Concept of Definite Integraland its Properties) (61)5.2 微积分基本定理(Fundamental Theorem of Calculus) (67)5.3 定积分的换元法和分部积分法(Integration by Substitution andDefinite Integrals by Parts) (69)5.4 反常积分(Improper Integrals) (70)第六章定积分的应用Chapter 6 Applications of the Definite Integrals (75)6.1 定积分的元素法(The Element Method of Definite Integra (75)6.2 定积分在几何学上的应用(Applications of the DefiniteIntegrals to Geometry) (76)6.3 定积分在物理学上的应用(Applications of the DefiniteIntegrals to Physics) (79)第七章空间解析几何与向量代数Chapter 7 Space Analytic Geometry and Vector Algebar (80)7.1 向量及其线性运算(Vector and Its Linear Operation) (80)7.2 数量积向量积(Dot Product and Cross Product) (86)7.3 曲面及其方程(Surface and Its Equation) (89)7.4 空间曲线及其方程(The Curve in Three-space and Its Equation (91)7.5 平面及其方程（Plane in Space and Its Equation) (93)7.6 空间直线及其方程（Lines in and Their Equations) (95)第八章多元函数微分法及其应用Chapter 8 Differentiation of Functions of SeveralVariables and Its Application (99)8.1 多元函数的基本概念（The Basic Concepts of Functionsof Several Variables) (99)8.2 偏导数(Partial Derivative) (102)8.3 全微分(Total Differential) (103)8.4 链式法则（The Chain Rule) (104)8.5 隐函数的求导公式(Derivative Formula for Implicit Functions). (104)8.6 多元函数微分学的几何应用(Geometric Applications of Differentiationof Ffunctions of Severalvariables) (106)8.7方向导数与梯度(Directional Derivatives and Gradients) (107)8.8多元函数的极值(Extreme Value of Functions of Several Variables) (108)第九章重积分Chapter 9 Multiple Integrals (111)9.1二重积分的概念与性质(The Concept of Double Integralsand Its Properities) (111)9.2二重积分的计算法(Evaluation of double Integrals) (114)9.3三重积分(Triple Integrals) (115)9.4重积分的应用(Applications of Multiple Itegrals) (120)第十章曲线积分与曲面积分Chapte 10 Line Integrals and Surface Integrals (121)10.1 对弧长的曲线积分(line Intergrals with Respect to Arc Length) (121)10.2 对坐标的曲线积分(Line Integrals with respect toCoordinate Variables) (123)10.3 格林公式及其应用(Green's Formula and Its Applications) (124)10.4 对面积的曲面积分(Surface Integrals with Respect to Aarea) (126)10.5 对坐标的曲面积分(Surface Integrals with Respect toCoordinate Variables) (128)10.6 高斯公式通量与散度(Gauss's Formula Flux and Divirgence) (130)10.7 斯托克斯公式环流量与旋度(Stokes's Formula Circulationand Rotation) (131)第十一章无穷级数Chapter 11 Infinite Series (133)11.1 常数项级数的概念与性质(The concept and Properties ofThe Constant series) (133)11.2 常数项级数的审敛法(Test for Convergence of the Constant Series) (137)11.3 幂级数(power Series). (143)11.4 函数展开成幂级数(Represent the Function as Power Series) (148)11.5 函数的幂级数展开式的应用(the Appliacation of the Power Seriesrepresentation of a Function) (148)11.6 函数项级数的一致收敛性及一致收敛级数的基本性质(The UnanimousConvergence of the Series of Functions and Its properties) (149)11.7 傅立叶级数（Fourier Series） (152)11.8 一般周期函数的傅立叶级数(Fourier Series of Periodic Functions) (153)第十二章微分方程Chapter 12 Differential Equation (155)12.1微分方程的基本概念（The Concept of DifferentialEquation) (155)12.2可分离变量的微分方程(Separable Differential Equation) (156)12.3齐次方程(Homogeneous Equation) (156)12.4 一次线性微分方程(Linear Differential Equation of theFirst Order) (157)12.5全微分方程(Total Differential Equation) (158)12.6可降阶的高阶微分方程(Higher-order DifferentialEquation Turned to Lower-order DifferentialEquation) (159)12.7高阶线性微分方程(Linear Differential Equation of HigherOrder) (159)12.8常系数齐次线性微分方程(Homogeneous LinearDifferential Equation with Constant Coefficient) (163)12.9常系数非齐次线性微分方程(Non HomogeneousDifferential Equation with Constant Coefficient) (164)12.10 欧拉方程(Euler Equation) (164)12.11 微分方程的幂级数解法(Power Series Solutionto Differential Equation) (164)第三部分常用数学符号的英文表达Part 3 English Expression of the Mathematical Symbol in Common Use第一部分英汉微积分词汇Part1 English-Chinese Calculus V ocabulary 第一章函数与极限Chapter1 Function and Limit集合set元素element子集subset空集empty set并集union交集intersection差集difference of set基本集basic set补集complement set直积direct product笛卡儿积Cartesian product开区间open interval闭区间closed interval半开区间half open interval有限区间finite interval区间的长度length of an interval无限区间infinite interval领域neighborhood领域的中心centre of a neighborhood领域的半径radius of a neighborhood左领域left neighborhood右领域right neighborhood 映射mappingX到Y的映射mapping of X ontoY 满射surjection单射injection一一映射one-to-one mapping双射bijection算子operator变化transformation函数function逆映射inverse mapping复合映射composite mapping自变量independent variable因变量dependent variable定义域domain函数值value of function函数关系function relation值域range自然定义域natural domain单值函数single valued function多值函数multiple valued function 单值分支one-valued branch函数图形graph of a function绝对值函数absolute value符号函数sigh function整数部分integral part阶梯曲线step curve当且仅当if and only if(iff)分段函数piecewise function上界upper bound下界lower bound有界boundedness无界unbounded函数的单调性monotonicity of a function 单调增加的increasing单调减少的decreasing单调函数monotone function函数的奇偶性parity(odevity) of a function对称symmetry偶函数even function奇函数odd function函数的周期性periodicity of a function周期period反函数inverse function直接函数direct function复合函数composite function中间变量intermediate variable函数的运算operation of function基本初等函数basic elementary function初等函数elementary function幂函数power function指数函数exponential function对数函数logarithmic function三角函数trigonometric function反三角函数inverse trigonometric function 常数函数constant function双曲函数hyperbolic function双曲正弦hyperbolic sine双曲余弦hyperbolic cosine双曲正切hyperbolic tangent反双曲正弦inverse hyperbolic sine反双曲余弦inverse hyperbolic cosine反双曲正切inverse hyperbolic tangent极限limit数列sequence of number收敛convergence收敛于 a converge to a发散divergent极限的唯一性uniqueness of limits收敛数列的有界性boundedness of a convergent sequence子列subsequence函数的极限limits of functions函数()f x当x趋于x0时的极限limit of functions () f x as x approaches x0左极限left limit右极限right limit单侧极限one-sided limits水平渐近线horizontal asymptote无穷小infinitesimal无穷大infinity铅直渐近线vertical asymptote夹逼准则squeeze rule单调数列monotonic sequence高阶无穷小infinitesimal of higher order低阶无穷小infinitesimal of lower order同阶无穷小infinitesimal of the same order 等阶无穷小equivalent infinitesimal函数的连续性continuity of a function增量increment函数()f x在x0连续the function ()f x is continuous at x0左连续left continuous右连续right continuous区间上的连续函数continuous function函数()f x在该区间上连续function ()f x is continuous on an interval不连续点discontinuity point第一类间断点discontinuity point of the first kind第二类间断点discontinuity point of the second kind初等函数的连续性continuity of the elementary functions定义区间defined interval最大值global maximum value (absolute maximum)最小值global minimum value (absolute minimum)零点定理the zero point theorem介值定理intermediate value theorem第二章导数与微分Chapter2 Derivative and Differential速度velocity匀速运动uniform motion平均速度average velocity瞬时速度instantaneous velocity圆的切线tangent line of a circle切线tangent line切线的斜率slope of the tangent line位置函数position function导数derivative可导derivable函数的变化率问题problem of the change rate of a function 导函数derived function左导数left-hand derivative右导数right-hand derivative单侧导数one-sided derivatives()f x在闭区间【a,b】上可导()f x is derivable on the closed interval [a,b]切线方程tangent equation角速度angular velocity成本函数cost function边际成本marginal cost链式法则chain rule隐函数implicit function显函数explicit function二阶函数second derivative三阶导数third derivative高阶导数nth derivative莱布尼茨公式Leibniz formula对数求导法log- derivative参数方程parametric equation相关变化率correlative change rata微分differential可微的differentiable函数的微分differential of function自变量的微分differential of independent variable微商differential quotient间接测量误差indirect measurement error 绝对误差absolute error 相对误差relative error第三章微分中值定理与导数的应用Chapter3 MeanValue Theorem of Differentials and the Application of Derivatives 罗马定理Rolle’s theorem费马引理Fermat’s lemma拉格朗日中值定理Lagrange’s mean value theorem驻点stationary point稳定点stable point临界点critical point辅助函数auxiliary function拉格朗日中值公式Lagrange’s mean value formula柯西中值定理Cauchy’s mean value theorem洛必达法则L’Hospital’s Rule0/0型不定式indeterminate form of type 0/0不定式indeterminate form泰勒中值定理Taylor’s mean value theorem泰勒公式Taylor formula余项remainder term拉格朗日余项Lagrange remainder term 麦克劳林公式Maclaurin’s formula佩亚诺公式Peano remainder term凹凸性concavity凹向上的concave upward, cancave up凹向下的，向上凸的concave downward’concave down拐点inflection point函数的极值extremum of function极大值local(relative) maximum最大值global(absolute) mximum极小值local(relative) minimum最小值global(absolute) minimum目标函数objective function曲率curvature弧微分arc differential平均曲率average curvature曲率园circle of curvature曲率中心center of curvature曲率半径radius of curvature渐屈线evolute渐伸线involute根的隔离isolation of root隔离区间isolation interval切线法tangent line method第四章不定积分Chapter4 Indefinite Integrals原函数primitive function(antiderivative) 积分号sign of integration被积函数integrand积分变量integral variable积分曲线integral curve积分表table of integrals换元积分法integration by substitution分部积分法integration by parts分部积分公式formula of integration by parts有理函数rational function真分式proper fraction假分式improper fraction第五章定积分Chapter5 Definite Integrals曲边梯形trapezoid with曲边curve edge窄矩形narrow rectangle曲边梯形的面积area of trapezoid with curved edge积分下限lower limit of integral积分上限upper limit of integral积分区间integral interval分割partition积分和integral sum可积integrable矩形法rectangle method积分中值定理mean value theorem of integrals函数在区间上的平均值average value of a function on an integvals牛顿－莱布尼茨公式Newton-Leibniz formula微积分基本公式fundamental formula of calculus换元公式formula for integration by substitution 递推公式recurrence formula反常积分improper integral反常积分发散the improper integral is divergent反常积分收敛the improper integral is convergent无穷限的反常积分improper integral on an infinite interval无界函数的反常积分improper integral of unbounded functions绝对收敛absolutely convergent第六章定积分的应用Chapter6 Applications of the Definite Integrals元素法the element method面积元素element of area平面图形的面积area of a luane figure直角坐标又称“笛卡儿坐标(Cartesian coordinates)”极坐标polar coordinates抛物线parabola椭圆ellipse旋转体的面积volume of a solid of rotation旋转椭球体ellipsoid of revolution, ellipsoid of rotation曲线的弧长arc length of acurve可求长的rectifiable光滑smooth功work水压力water pressure引力gravitation变力variable force第七章空间解析几何与向量代数Chapter7 Space Analytic Geometry and Vector Algebra向量vector自由向量free vector单位向量unit vector零向量zero vector相等equal平行parallel向量的线性运算linear poeration of vector 三角法则triangle rule平行四边形法则parallelogram rule交换律commutative law结合律associative law负向量negative vector差difference分配律distributive law空间直角坐标系space rectangular coordinates坐标面coordinate plane卦限octant向量的模modulus of vector向量a与b的夹角angle between vector a and b方向余弦direction cosine方向角direction angle向量在轴上的投影projection of a vector onto an axis数量积，外积，叉积scalar product,dot product,inner product 曲面方程equation for a surface球面sphere旋转曲面surface of revolution母线generating line轴axis圆锥面cone顶点vertex旋转单叶双曲面revolution hyperboloids of one sheet旋转双叶双曲面revolution hyperboloids of two sheets柱面cylindrical surface ,cylinder圆柱面cylindrical surface准线directrix抛物柱面parabolic cylinder二次曲面quadric surface椭圆锥面dlliptic cone椭球面ellipsoid单叶双曲面hyperboloid of one sheet双叶双曲面hyperboloid of two sheets旋转椭球面ellipsoid of revolution椭圆抛物面elliptic paraboloid旋转抛物面paraboloid of revolution双曲抛物面hyperbolic paraboloid马鞍面saddle surface 椭圆柱面elliptic cylinder双曲柱面hyperbolic cylinder抛物柱面parabolic cylinder空间曲线space curve空间曲线的一般方程general form equations of a space curve 空间曲线的参数方程parametric equations of a space curve螺转线spiral螺矩pitch投影柱面projecting cylinder投影projection平面的点法式方程pointnorm form eqyation of a plane法向量normal vector平面的一般方程general form equation of a plane两平面的夹角angle between two planes 点到平面的距离distance from a point to a plane空间直线的一般方程general equation of a line in space方向向量direction vector直线的点向式方程pointdirection form equations of a line方向数direction number直线的参数方程parametric equations of a line两直线的夹角angle between two lines垂直perpendicular直线与平面的夹角angle between a line and a planes平面束pencil of planes平面束的方程equation of a pencil of planes行列式determinant系数行列式coefficient determinant第八章多元函数微分法及其应用Chapter8 Differentiation of Functions of Several Variables and Its Application一元函数function of one variable多元函数function of several variables内点interior point外点exterior point边界点frontier point,boundary point聚点point of accumulation开集openset闭集closed set连通集connected set开区域open region闭区域closed region有界集bounded set无界集unbounded setn维空间n-dimentional space二重极限double limit多元函数的连续性continuity of function of seveal连续函数continuous function不连续点discontinuity point一致连续uniformly continuous偏导数partial derivative对自变量x的偏导数partial derivative with respect to independent variable x高阶偏导数partial derivative of higher order二阶偏导数second order partial derivative 混合偏导数hybrid partial derivative全微分total differential偏增量oartial increment偏微分partial differential全增量total increment可微分differentiable必要条件necessary condition充分条件sufficient condition叠加原理superpostition principle全导数total derivative中间变量intermediate variable隐函数存在定理theorem of the existence of implicit function 曲线的切向量tangent vector of a curve法平面normal plane向量方程vector equation向量值函数vector-valued function切平面tangent plane法线normal line方向导数directional derivative梯度gradient 数量场scalar field梯度场gradient field向量场vector field势场potential field引力场gravitational field引力势gravitational potential曲面在一点的切平面tangent plane to a surface at a point曲线在一点的法线normal line to a surface at a point无条件极值unconditional extreme values 条件极值conditional extreme values拉格朗日乘数法Lagrange multiplier method拉格朗日乘子Lagrange multiplier经验公式empirical formula最小二乘法method of least squares均方误差mean square error第九章重积分Chapter9 Multiple Integrals二重积分double integral可加性additivity累次积分iterated integral体积元素volume element三重积分triple integral直角坐标系中的体积元素volume element in rectangular coordinate system柱面坐标cylindrical coordinates柱面坐标系中的体积元素volume element in cylindrical coordinate system球面坐标spherical coordinates球面坐标系中的体积元素volume element in spherical coordinate system反常二重积分improper double integral曲面的面积area of a surface质心centre of mass静矩static moment密度density形心centroid转动惯量moment of inertia参变量parametric variable第十章曲线积分与曲面积分Chapter10 Line(Curve)Integrals and Surface Integrals对弧长的曲线积分line integrals with respect to arc hength第一类曲线积分line integrals of the first type对坐标的曲线积分line integrals with respect to x,y,and z第二类曲线积分line integrals of the second type有向曲线弧directed arc单连通区域simple connected region复连通区域complex connected region格林公式Green formula第一类曲面积分surface integrals of the first type对面的曲面积分surface integrals with respect to area有向曲面directed surface对坐标的曲面积分surface integrals with respect to coordinate elements第二类曲面积分surface integrals of the second type有向曲面元element of directed surface高斯公式gauss formula拉普拉斯算子Laplace operator格林第一公式Green’s first formula通量flux散度divergence斯托克斯公式Stokes formula环流量circulation旋度rotation,curl第十一章无穷级数Chapter11 Infinite Series一般项general term部分和partial sum余项remainder term等比级数geometric series几何级数geometric series公比common ratio调和级数harmonic series柯西收敛准则Cauchy convergence criteria, Cauchy criteria for convergence正项级数series of positive terms达朗贝尔判别法D’Alembert test柯西判别法Cauchy test 交错级数alternating series绝对收敛absolutely convergent条件收敛conditionally convergent柯西乘积Cauchy product函数项级数series of functions发散点point of divergence收敛点point of convergence收敛域convergence domain和函数sum function幂级数power series幂级数的系数coeffcients of power series 阿贝尔定理Abel Theorem收敛半径radius of convergence收敛区间interval of convergence泰勒级数Taylor series麦克劳林级数Maclaurin series二项展开式binomial expansion近似计算approximate calculation舍入误差round-off error,rounding error欧拉公式Euler’s formula魏尔斯特拉丝判别法Weierstrass test三角级数trigonometric series振幅amplitude角频率angular frequency初相initial phase矩形波square wave谐波分析harmonic analysis直流分量direct component基波fundamental wave二次谐波second harmonic三角函数系trigonometric function system 傅立叶系数Fourier coefficient傅立叶级数Forrier series周期延拓periodic prolongation正弦级数sine series余弦级数cosine series奇延拓odd prolongation偶延拓even prolongation傅立叶级数的复数形式complex form of Fourier series第十二章微分方程Chapter12 Differential Equation解微分方程solve a dirrerential equation 常微分方程ordinary differential equation偏微分方程partial differential equation,PDE微分方程的阶order of a differential equation微分方程的解solution of a differential equation微分方程的通解general solution of a differential equation初始条件initial condition微分方程的特解particular solution of a differential equation 初值问题initial value problem微分方程的积分曲线integral curve of a differential equation 可分离变量的微分方程variable separable differential equation 隐式解implicit solution隐式通解inplicit general solution衰变系数decay coefficient衰变decay齐次方程homogeneous equation一阶线性方程linear differential equation of first order非齐次non-homogeneous齐次线性方程homogeneous linear equation非齐次线性方程non-homogeneous linear equation常数变易法method of variation of constant暂态电流transient stata current稳态电流steady state current伯努利方程Bernoulli equation全微分方程total differential equation积分因子integrating factor高阶微分方程differential equation of higher order悬链线catenary高阶线性微分方程linera differential equation of higher order 自由振动的微分方程differential equation of free vibration强迫振动的微分方程differential equation of forced oscillation 串联电路的振荡方程oscillation equation of series circuit二阶线性微分方程second order linera differential equation线性相关linearly dependence线性无关linearly independce二阶常系数齐次线性微分方程second order homogeneour linear differential equation with constant coefficient二阶变系数齐次线性微分方程second order homogeneous linear differential equation with variable coefficient特征方程characteristic equation无阻尼自由振动的微分方程differential equation of free vibration with zero damping 固有频率natural frequency 简谐振动simple harmonic oscillation,simple harmonic vibration微分算子differential operator待定系数法method of undetermined coefficient共振现象resonance phenomenon欧拉方程Euler equation幂级数解法power series solution数值解法numerial solution勒让德方程Legendre equation微分方程组system of differential equations常系数线性微分方程组system of linera differential equations with constant coefficient第二部分定理定义公式的英文表达Part2 English Expression for Theorem, Definition and Formula第一章函数与极限Chapter 1 Function and Limit1.1 映射与函数 (Mapping and Function)一、集合 (Set)二、映射 (Mapping)映射概念 (The Concept of Mapping) 设X , Y 是两个非空集合 , 如果存在一个法则f ，使得对X 中每个元素x ，按法则f ，在Y 中有唯一确定的元素y 与之对应 , 则称f 为从X 到 Y 的映射 , 记作:f X Y →。
法国数学家拉格朗日著作《解析函数论》英文名
法国数学家拉格朗日著作《解析函数论》英文名全文共3篇示例，供读者参考篇1Title: French Mathematician Lagrange's Work "Analytical Function Theory"Introduction:Lagrange's "Analytical Function Theory" is a seminal work by the French mathematician Joseph-Louis Lagrange, also known as the Lagrange interpolation or Lagrange polynomial. In this work, Lagrange presents a detailed analysis of functions and their properties, laying the foundation for modern function theory. The book delves into topics such as series, limits, derivatives, and integrals of functions, providing a comprehensive study of mathematical functions.Chapter 1: Historical BackgroundJoseph-Louis Lagrange was born in Turin, Italy, in 1736 and later moved to Paris, where he made significant contributions to mathematics, mechanics, and astronomy. Lagrange's work in function theory was influenced by earlier mathematicians such as Euler, d'Alembert, and Legendre. His innovative approach toanalyzing functions set him apart as a pioneering figure in the field of mathematics.Chapter 2: Analytical Function TheoryIn "Analytical Function Theory," Lagrange explores the properties of functions through the use of calculus and algebraic techniques. He introduces the concept of series as a way to represent functions as infinite sums of terms, allowing for a more precise analysis of their behavior. Lagrange also discusses the importance of limits in determining the behavior of functions at particular points, laying the groundwork for modern calculus.Chapter 3: Applications and ImpactLagrange's work on function theory has had a lasting impact on mathematics, with his ideas forming the basis for modern theories in analysis and calculus. The concept of the Lagrange interpolation polynomial, named in his honor, is still widely used in numerical analysis and approximation techniques. His work has inspired generations of mathematicians to further explore the depths of function theory and its applications in various fields.Conclusion:In conclusion, Joseph-Louis Lagrange's work "Analytical Function Theory" stands as a testament to his brilliance and innovative contributions to the field of mathematics. Through his meticulous analysis of functions and their properties, Lagrange paved the way for future developments in function theory and calculus. His work remains a cornerstone of modern mathematics, continuing to inspire mathematicians worldwide to push the boundaries of knowledge in this dynamic field.篇2Title: French Mathematician Lagrange's Work "Analytic Functions Theory"IntroductionLagrange's work on Analytic Functions Theory is a significant contribution to the field of mathematics. This book, also known as "Théorie des Fonctions Analytiques" in French, was published in the year 1797 by the renowned French mathematician Joseph Louis Lagrange. In this seminal work, Lagrange established the foundation for the study of analytic functions and laid the groundwork for the development of complex analysis.Background of LagrangeJoseph Louis Lagrange, born in Turin, Italy in 1736, was a prominent mathematician who made groundbreaking contributions to various fields of mathematics, such as number theory, calculus, and celestial mechanics. He is widely regarded as one of the greatest mathematicians of all time and his work continues to influence modern mathematics.Content of the BookIn "Analytic Functions Theory", Lagrange delves into the study of functions that can be represented by a power series expansion. He introduces key concepts and theorems related to complex analysis, such as Cauchy's integral theorem, the residue theorem, and the fundamental theorem of algebra. Lagrange's work on the properties and behavior of analytic functions revolutionized the field of mathematics and paved the way for further developments in the study of complex numbers.Significance of the WorkLagrange's book is considered a seminal work in the field of mathematics and remains a cornerstone of complex analysis. His contributions to analytic functions theory have had a lasting impact on the field of mathematics and continue to influence contemporary research in areas such as number theory, physics,and engineering. The book has been widely studied and referenced by mathematicians and scientists around the world.ConclusionIn conclusion, Lagrange's work on Analytic Functions Theory stands as a testament to his brilliance and innovation in the field of mathematics. His insights and discoveries continue to shape our understanding of complex analysis and pave the way for new advancements in the field. The book remains a timeless classic in the realm of mathematics and serves as a source of inspiration for generations of mathematicians to come.篇3Title: The Analytic Function Theory by French Mathematician LagrangeIntroduction:Joseph-Louis Lagrange, a renowned mathematician from France, made significant contributions to the field of mathematics during the 18th century. One of his most influential works is the book "Analytic Function Theory," where he laid down the foundations for the study of complex functions. In this article, we will delve into the contents of this seminal work and discuss its impact on the development of mathematics.Overview of the Book:Lagrange's "Analytic Function Theory" is a comprehensive treatise on the analysis of complex functions, which play a crucial role in a variety of mathematical disciplines including calculus, differential equations, and number theory. The book is divided into several sections, each covering different aspects of the theory of analytic functions. Lagrange begins by introducing the basic concepts of complex numbers and functions, before delving into more advanced topics such as power series, contour integration, and the Cauchy-Riemann equations.Key Concepts and Theorems:One of the key contributions of Lagrange in this work is the development of the Cauchy Integral Formula, which provides a powerful method for calculating complex integrals over closed curves. This formula has important applications in the study of harmonic functions and the theory of residues. Lagrange also proved several important theorems in the book, including the Maximum Modulus Principle and the Riemann Mapping Theorem, which have been instrumental in the development of complex analysis.Impact on Mathematics:Lagrange's "Analytic Function Theory" is considered a seminal work in the field of complex analysis and has had a lasting impact on the development of mathematics. The insights and techniques introduced by Lagrange in this book have been instrumental in solving many mathematical problems in diverse areas such as physics, engineering, and computer science. The book continues to be studied and referenced by mathematicians and researchers around the world, highlighting the enduring legacy of Lagrange's contributions to the field.Conclusion:In conclusion, Joseph-Louis Lagrange's "Analytic Function Theory" stands as a cornerstone in the field of complex analysis, providing a solid foundation for the study of analytic functions and their applications. The insights and theorems introduced by Lagrange in this book have had a profound impact on the development of mathematics, shaping the way we understand and solve complex mathematical problems. As we continue to push the boundaries of mathematical research, Lagrange's work remains as relevant and influential as ever.。
人教版高中数学目录(中英对照版)
必修1Required Course 1第一章集合与函数概念Chapter 1 concepts of Set and Function1．1 集合1.1 Set1．2 函数及其表示1.2 Function and its expression1．3 函数的基本性质1.3 Basic properties of function第二章基本初等函数（Ⅰ）Chapter 2 Basic Elementary Functions (I)2．1 指数函数2.1 Exponential function2．2 对数函数2.2 Logarithmic function2．3 幂函数2.3 Power function第三章函数的应用Chapter 3 Application of function 3．1 函数与方程3.1 Function and equation3．2 函数模型及其应用3.2 Application of function model必修2Required Course 2第一章空间几何体Chapter 1 The space geometry1．1 空间几何体的结构1.1 Structure1．2 空间几何体的三视图和直观图1.2 Three views and intuitionistic figure 1．3 空间几何体的表面积与体积1.3 Surface area and volume第二章点、直线、平面之间的位置关系Chapter 2 Relational Positions of Point、line and plane 2．1 空间点、直线、平面之间的位置关系2.1 Relational Positions of Point、line and plane2．2 直线、平面平行的判定及其性质2.2 Determine and basic properties of Parallel lines and plane2．3 直线、平面垂直的判定及其性质2.3 Determine and basic properties of vertical and plane第三章直线与方程Chapter 3 Line and equation3．1 直线的倾斜角与斜率3.1 Angle of slope of a line3．2 直线的方程3.2 Linear equation3．3 直线的交点坐标与距离公式3.3 The intersection point coordinate and distance formula of a line必修3Required Course 3第一章算法初步Chapter 1 Algorithm preliminary 1．1 算法与程序框图1.1 Algorithm and program chart1．2 基本算法语句1.2 Basic algorithmic statements1．3 算法案例1.3 Algorithmic case第二章统计Chapter 2 Statistics2．1 随机抽样2.1 Random sampling2．2 用样本估计总体2.2 Overall estimate with sample2．3 变量间的相关关系2.3 Relationship between the variables第三章概率Chapter 3 Probability3．1 随机事件的概率3.1 Probability of random events3．2 古典概型3.2 Classical probability3．3 几何概型3.3 Geometric probability必修4Required Course 4第一章三角函数Chapter 1 Trigonometric function 1．1 任意角和弧度制1.1 Arbitrary angle and radian measure 1．2 任意角的三角函数1.2 Trigonometric function of arbitrary angle1．3 三角函数的诱导公式1.3 Induction formula of trigonometric function1．4 三角函数的图象与性质1.4 The image and properties of trigonometric function1．5 函数y=Asin（ωx+ψ）1.5 Function y=Asin（ωx+ψ）1．6 三角函数模型的简单应用1.6 Simple application of trigonometric function model第二章平面向量Chapter 2 Plane vector2．1 平面向量的实际背景及基本概念2.1 Background and concept2．2 平面向量的线性运算2.2 Linear operation2．3 平面向量的基本定理及坐标表示2.3 The Fundamental Theorem and coordinate representation2．4 平面向量的数量积2.4 Dot Product2．5 平面向量应用举例2.5 Application Example 第三章三角恒等变换Chapter 3 Triangle identical transformation3．1 两角和与差的正弦、余弦和正切公式3.1 Concepts of sine、cosine and tangent 3．2 简单的三角恒等变换3.2 Simple triangle identical transformation必修5Required Course 5第一章解三角形Chapter 1 Solving triangles1.1正弦定理和余弦定理1.1 Laws of sine and cosine1.2应用举例1.2 Application Example1.3实习作业1.3 Training assignment第二章数列Chapter 2 Sequences2.1数列的概念与简单表示法2.1 Concept and simple notation of Sequences2.2等差数列2.2 Arithmetic progression2.3等差数列的前n项和2.3 The sum of first N terms2.4等比数列2.4 Geometric progression2.5等比数列的前n项和2.5 The sum of first N terms第三章不等式Chapter 3 Inequality3.1不等关系与不等式3.1 Relation of inequality and inequality 3.2一元二次不等式及其解法3.2 One-variable quadratic inequality and its solution3.3二元一次不等式（组）与简单的线性规划问题3.3 One-variable quadratic inequality and simple linear programming problem 3.4基本不等式3.4 Fundamental inequality选修2-1Elective Course 2-1第一章常用逻辑用语Chapter 1 Logic language1.1命题及其关系1.1 Proposition and its relation1.2充分条件与必要条件1.2 Sufficient condition and necessary condition1.3简单的逻辑联结词1.3 Logical Connectives1.4全称量词与存在量词1.4 Universal and existential quantification第二章圆锥曲线与方程Chapter 2 Conic section and equation 2.1曲线与方程2.1 Curve and equation2.2椭圆2.2 Oval2.3双曲线2.3 Hyperbola2.4抛物线2.4 Parabola第三章空间向量与立体几何Chapter 3 Space vector and solid geometry3.1空间向量及其运算3.1 Space vector and operation 3.2立体几何中的向量方法3.2 vector methods in solid geometry选修2-2Elective Course 2-2第一章导数及其应用Chapter 1 Derivative and application 1.1变化率与导数1.1 Rate of Change and derivative1.2导数的计算1.2 The calculation of the derivative1.3导数在研究函数中的应用1.3 Application of derivative in function study1.4生活中的优化问题举例1.4 Optimization problem in daily life 1.5定积分的概念1.5 Concept of definite integral1.6微积分基本定理1.6 The Fundamental Theorem of Calculus1.7定积分的简单应用1.7 Application of definite integral第二章推理与证明Chapter 2 Reasoning and proof2.1合情推理与演绎推理2.1 Plausible reasoning and deductive reasoning2.2直接证明与间接证明2.2 Direct proof and indirect proof2.3数学归纳法2.3 Mathematical induction第三章数系的扩充与复数的引入Chapter 3 Expansion of number systems and complex number3.1数系的扩充和复数的概念3.1 Expansion of number systems and the concept of complex number3.2复数代数形式的四则运算3.2 Elementary arithmetic of complex number in algebraic forms选修2-3Elective Course 2-3第一章计数原理Chapter 1 Principle of counting1.1分类加法计数原理与分步乘法计数原理1.1 Counting principles of addition and multiplication1.2排列与组合1.2 Permutation and Combination1.3二项式定理1.3 Binomial theorem第二章随机变量及其分布Chapter 2 Random variable and distribution2.1离散型随机变量及其分布列2.1 Discrete random variables and distribution2.2二项分布及其应用2.2 Binominal distribution and application2.3离散型随机变量的均值与方差2.3 The mean and variance of discrete random variables2.4正态分布2.4 Normal distribution第三章统计案例Chapter 3 Statistical samples3.1回归分析的基本思想及其初步应用3.1 Basic thought and primary application of regression analysis3.2独立性检验的基本思想及其初步应用3.2 Basic thought and primary application of independence test选修4-1Elective Course 4-1第一讲相似三角形的判定及有关性质1. Determine and properties of similar triangles第二讲直线与圆的位置关系2. Relational positions of Straight line and circle第三讲圆锥曲线性质的探讨3. Properties of a curve of a circular cone选修4-4Elective Course 4-4第一讲坐标系1.Coordinate system第二讲参数方程2. Parametric Equation选修4-5Elective Course 4-5第一讲不等式和绝对值不等式1.Inequality and inequality with absolute value第二讲证明不等式的基本方法2. common methods about the demonstration of Inequality第三讲柯西不等式与排序不等式3. Cauchy inequality and rearrangement inequality第四讲数学归纳法证明不等式4. demonstration of Inequality with mathematical induction。
上海财经大学英语高数课件05
Example 2: Find the area under the parabola y=x2+1 from 0 to 2.
Solution Since y=x2+1 is continuous, the limit (1) must exist for all possible partition P of the interval [a, b] as long as ||P|| 0. To simplify things let us take a regular partition. Then the partition points are x0=0, x1=2/n, x2=4/n, … , xi=2i/n, … , xn=2n/n=2
a
f ( x)dx lim
|| P|| 0
f ( x i )xi i 1
n
if this limit exists. If the limit does exist, then f is called integrable on the interval [a, b]. Note 1:
So the norm of P is
||P||=2/n Let us choose the point xi to be the right-hand endpoint:
xi = xi=2i/n
By definition, the area is
机动 目录 上页 下页 返回 结束
i 2 14 A lim f ( xi )xi lim f ( 2 ) n n 3 || P|| 0 i 1 n i 1
sin b
/section 5.2 end
微积分第五章课件
▪ 5.1 Areas and Distances
The area problem
Area of rectangle Area of trapezoid 1. Areas of curved trapezoid Suppose the curved trapezoid is bounded by
Next we assume that f(x) is not a constant function. Since f(x) is continuous on [a, b], f(x) takes on the minimum and the maximum values on [a, b]. Let f(u) = m and f(v) = M be the
1 0
x2
dx
lim
0
n
i 1
i
2xi
y
y x2
lim
n
1 3
o
i 1x
n
Example 2. Use the definition to evaluate Solution: Use
to divide [1, 3] into n subintervals of equal width. We choose
and Find the area of A .
y f (x)
A?
Method:
1) Partition:
a x0 x1 x2 xn1 xn b
Use the lines x xi to divide A into small curved trapezoid;
2) Approximation: i [xi1 , xi ]
letter in place of x without changing the value of
大一微积分知识点英文
大一微积分知识点英文Calculus Knowledge Points for FreshmenCalculus, a fundamental branch of mathematics, is essential for students majoring in science, engineering, and mathematics. Mastering the core concepts and principles of calculus is crucial for a successful academic journey in these fields. In this article, we will explore some key calculus knowledge points for freshmen.1. Limits:Limits are fundamental to the study of calculus. A limit represents the value that a function or sequence approaches as its input or index approaches a certain point. Limits are extensively used to define derivatives and integrals.2. Derivatives:Derivatives measure the rate at which a function changes. It represents the slope of the tangent line to a curve at a particular point. Derivatives allow us to analyze the behavior of functions, determine critical points, and solve optimization problems. Notation for derivatives includes the prime symbol (') and the differential operator d/dx.3. Differentiation Rules:Differentiation rules provide shortcuts for computing derivatives. Some of the important rules include the power rule, product rule, quotient rule, chain rule, and trigonometric derivatives. Understanding these rules simplifies the process of finding derivatives of functions.4. Applications of Derivatives:Derivatives have various applications in real-life scenarios. They can be used to determine velocity and acceleration, solve related rates problems, find maximum and minimum values, and analyze the behavior of functions. Application areas include physics, economics, engineering, and biology.5. Integrals:Integrals, also known as antiderivatives, are the reverse process of derivatives. They represent the accumulation of quantities over an interval. Integrals are used to find areas, volumes, average values, and solve differential equations. Notation for integrals includes the integral symbol (∫) and the differential operator dx.6. Integration Techniques:Integration techniques provide methods for computing integrals. These techniques include u-substitution, integration by parts, trigonometric substitutions, and partial fractions. Mastery of these techniques enables students to evaluate a wide range of integrals efficiently.7. Applications of Integrals:Integrals have numerous applications, particularly in calculating areas and volumes. They can be used to find the area between curves, volumes of solids of revolution, work done by a force, and average values of functions. Integration is a powerful tool in physics, engineering, and economics.8. Fundamental Theorem of Calculus:The Fundamental Theorem of Calculus establishes the relationship between differentiation and integration. It states that the derivative of an integral of a function is equal to the original function. This theorem allows for the evaluation of definite integrals using antiderivatives.9. Sequences and Series:Sequences and series involve the summation of infinite terms. Convergence and divergence of sequences and series are crucialconcepts in calculus. Tests such as the ratio test, comparison test, and integral test can determine the convergence or divergence of a series.10. Multivariable Calculus:Multivariable calculus extends the concepts of calculus to functions of multiple variables. It involves partial derivatives, gradient vectors, multiple integrals, line integrals, and surface integrals. Multivariable calculus is essential for fields such as physics, computer science, and engineering.In summary, these calculus knowledge points provide a foundation for freshmen to embark on their study in calculus. Understanding and applying these concepts will enable students to solve complex problems and analyze real-world phenomena. By building a solid understanding of calculus, students can pave the way for success in their academic and professional pursuits.。
牛顿莱布尼茨公式求导
牛顿莱布尼茨公式求导牛顿-莱布尼茨公式（Fundamental Theorem of Calculus）是微积分中十分重要的定理，它可用于求导和不定积分之间的关系。
公式的完整形式如下：设函数 f(x) 在闭区间 [a, b] 上连续，且在 (a, b) 内可导。
令 F(x) 为函数 f(x) 在区间 [a, x] 上的不定积分，则有：∫[a, b] f(x)dx = F(b) - F(a)牛顿-莱布尼茨公式提供了一种通过不定积分的计算来求解定积分的方法。
在该公式中，F(x) 是 f(x) 的原函数，即 F'(x) =f(x)。
换句话说，F(x) 的导数等于函数 f(x)。
通过计算函数 f(x) 的原函数F(x)，我们可以使用牛顿-莱布尼茨公式来求函数f(x) 在给定区间 [a, b] 上的定积分。
公式要求函数在闭区间 [a, b] 上连续，这是为了保证函数 f(x)在该区间上有定义。
同时，函数 f(x) 在开区间 (a, b) 内可导，则可以保证在区间内的每个点上都存在导数，从而满足原函数的存在性。
牛顿-莱布尼茨公式的应用十分广泛，许多微积分的问题都可以通过该公式解决。
例如，可以利用该公式计算函数在给定区间上的平均值、最大值和最小值，以及计算弧长、面积和体积等。
此外，该公式还可以用于解决微分方程和偏微分方程等数学问题。
下面以一个具体的例子来解释牛顿-莱布尼茨公式的应用。
考虑函数 f(x) = x²，在区间 [1, 2] 上求定积分∫[1, 2] x² dx。
我们可以首先求 f(x) 的原函数 F(x)，由于 F(x) 的导函数为 f(x)，所以 F(x) = (1/3)x³。
然后，将 F(2) 和 F(1) 代入计算公式：F(2) -F(1) = (1/3)(2³) - (1/3)(1³) = 8/3 - 1/3 = 7/3，即定积分的结果为7/3。
积分 英语 读法
积分英语读法The Language of the Universe" with more than 1000 words, written entirely in English:Integral Calculus: The Language of the UniverseIn the vast expanse of human knowledge, few concepts have had as profound an impact on our understanding of the world as integral calculus. This branch of mathematics, with its intricate web of concepts and applications, has become the lingua franca of the scientific community, a universal language that allows us to decipher the intricacies of the natural world.At its core, integral calculus is concerned with the accumulation of infinitesimal quantities, a process that allows us to measure and quantify the behavior of continuous phenomena. From the graceful curves of a suspension bridge to the pulsing rhythm of the human heart, integral calculus provides the mathematical framework to unravel the mysteries of the universe.One of the most remarkable aspects of integral calculus is its ability to capture the essence of change. By considering the infinitesimalchanges that occur within a system, we can gain a deeper understanding of the larger trends and patterns that govern the world around us. This concept, known as the fundamental theorem of calculus, lies at the heart of integral calculus, connecting the rate of change (the derivative) with the accumulation of change (the integral).This powerful connection between the derivative and the integral has far-reaching implications, allowing us to model and predict a wide range of phenomena, from the motion of celestial bodies to the flow of fluids through a pipe. By understanding the relationship between these two fundamental concepts, we can unlock the secrets of the natural world, revealing the underlying patterns and principles that govern its behavior.But the true beauty of integral calculus lies not only in its mathematical elegance but also in its ability to transcend the boundaries of disciplines. From physics and engineering to economics and biology, this versatile tool has become an indispensable part of the scientific toolkit, enabling researchers to tackle the most complex problems with precision and rigor.In the realm of physics, for example, integral calculus is the foundation upon which the laws of motion and the principles of thermodynamics are built. By integrating the rate of change of aphysical quantity, such as position or energy, we can determine the overall behavior of a system, whether it's a falling object or a heat engine. This powerful approach has allowed us to unravel the mysteries of the universe, from the smallest subatomic particles to the grandest celestial bodies.Similarly, in engineering, integral calculus is the key to designing and optimizing complex systems, from the aerodynamics of aircraft to the efficiency of power plants. By modeling the flow of fluids, the distribution of stresses, and the accumulation of energy, engineers can create structures and machines that push the boundaries of what is possible.Even in the social sciences, integral calculus has found a home, allowing economists to analyze the flow of money, the accumulation of wealth, and the dynamics of market trends. By integrating the rate of change of economic variables, such as supply and demand, researchers can develop models that help us understand the complex systems that govern our economic landscape.But the true power of integral calculus lies not only in its practical applications but also in its ability to inspire wonder and awe. As we delve deeper into the mysteries of the universe, we find that the language of mathematics, and in particular the language of integral calculus, is the key to unlocking the secrets of the natural world.Consider, for example, the breathtaking beauty of a rainbow. This natural phenomenon, the result of the refraction and dispersion of sunlight through water droplets in the atmosphere, can be described and understood through the principles of integral calculus. By integrating the rate of change of the angle of refraction, we can predict the precise colors and patterns that will appear in the sky, a testament to the elegance and power of this mathematical tool.Or take the intricate patterns of a snowflake, each one a unique and exquisite work of art. These delicate structures, formed by the accumulation of water molecules in the atmosphere, can be modeled and understood through the principles of integral calculus. By integrating the rate of change of the temperature and humidity in the air, we can begin to unravel the complex processes that give rise to these beautiful and ephemeral creations.In the end, integral calculus is not just a mathematical tool but a language, a way of understanding the world that transcends the boundaries of discipline and culture. It is a language that allows us to communicate the mysteries of the universe, to unlock the secrets of the natural world, and to push the boundaries of what is possible.As we continue to explore the frontiers of human knowledge, it is clear that integral calculus will remain a vital and indispensable partof our toolkit. Whether we are studying the motion of planets, the flow of fluids, or the dynamics of the economy, this powerful mathematical language will continue to guide us, inspiring us to delve deeper into the wonders of the universe and to uncover the hidden patterns that govern our world.。
实系数多项式因式分解定理 英文
实系数多项式因式分解定理英文English:"The Fundamental Theorem of Algebra states that every non-constant polynomial equation with complex coefficients has at least one complex root. This implies that every polynomial of degree n can be factored into n linear factors over the complex numbers, where each factor corresponds to a root of the polynomial. In other words,if P(z) is a polynomial of degree n, then it can be expressed as P(z) = a(z - z1)(z - z2)...(z - zn), where a is the leading coefficient, z1, z2,...,zn are the roots of the polynomial, and n is the degree of the polynomial. This theorem provides a fundamental understanding of the behavior of polynomial equations and is crucial in various areasof mathematics and its applications, such as algebra, calculus, and engineering."中文翻译:"代数基本定理表明，具有复系数的每个非常数多项式方程至少有一个复根。
英语微积分知识点总结
英语微积分知识点总结Limits are a fundamental concept in calculus, and they are used to describe the behavior of a function as the input approaches a certain value. The limit of a function f(x) as x approaches a certain value a, denoted by lim(x→a) f(x), describes the value that f(x) approaches as x gets closer and closer to a. For example, the limit of the function f(x) = x^2 as x approaches 3 is 9, because as x gets closer and closer to 3, f(x) gets closer and closer to 9. Limits are used to define derivatives and integrals, which are the two central concepts in calculus.The derivative of a function f(x) describes how the function changes as its input changes. It is denoted by f'(x) or df/dx, and it represents the rate of change of the function at a particular point. The derivative of a function f(x) at a point x, denoted by f'(x) or df/dx, is defined as the limit of the average rate of change of f(x) as the interval around x shrinks to zero. Geometrically, the derivative represents the slope of the tangent line to the graph of the function at a particular point. It has many important applications in fields such as physics, engineering, and economics, where it is used to model and analyze rates of change of various quantities.The integral of a function f(x) describes the accumulation of the function over a certain interval. It is denoted by ∫f(x) dx, and it represents the area under the curve of the function over a certain interval. The integral of a function f(x) from a to b, denoted by ∫[a,b] f(x) dx, is defined as the limit of the sum of the areas of rectangles that overestimate and underestimate the area under the curve of f(x) as the width of the rectangles approaches zero. The integral has many important applications in fields such as physics, engineering, and economics, where it is used to calculate areas, volumes, and averages of various quantities.There are many techniques and methods for finding derivatives and integrals of functions. For derivatives, the most basic method is to use the definition of the derivative as a limit of a difference quotient. This method involves finding the limit of the average rate of change of the function as the interval around a point shrinks to zero. There are also rules and formulas for finding derivatives of specific functions, such as the power rule, the product rule, the quotient rule, and the chain rule. These rules and formulas can be used to find derivatives of more complex functions by applying them to simpler functions and using properties of limits.For integrals, the most basic method is to use the definition of the integral as a limit of a sum of areas of rectangles. This method involves dividing the interval over which the integral is being calculated into smaller and smaller subintervals, and then finding the sum of the areas of rectangles that overestimate and underestimate the area under the curve of the function on each subinterval. There are also rules and formulas for finding integrals of specific functions, such as the power rule for integrals, the substitution rule, and integration by parts. These rules and formulas can be used to find integrals of more complex functions by applying them to simpler functions and using properties of limits.One important property of integrals is the fundamental theorem of calculus, which relates the concept of the integral to the concept of the derivative. The fundamental theorem of calculus states that if F(x) is the antiderivative of a function f(x), then the integral of f(x) from a to b is equal to the difference between the values of F(x) at b and a, i.e.,∫[a,b] f(x) dx = F(b) - F(a).This theorem allows us to use the concept of the integral to find antiderivatives of functions, and it provides a powerful tool for calculating areas and volumes of various quantities.In conclusion, calculus is a fundamental tool in mathematics and the sciences, and it allows us to understand and analyze patterns of change in the world around us. Key concepts and principles of calculus include limits, derivatives, and integrals, and there are many techniques and methods for finding derivatives and integrals of functions. The fundamental theorem of calculus relates the concept of the integral to the concept of the derivative, and it provides a powerful tool for calculating areas and volumes of various quantities. Calculus has many important applications in fields such as physics, engineering, and economics, and it is a central part of the modern mathematical and scientific toolkit.。
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P (x)dx
= (165x − 0.05x2 )|1600 1500 = 1000 . Thus, increasing monthly production from 1,500 units to 1,600 units will increase the monthly profit by $1,000. (See College Mathematics for Business, §12-5, Example 5.)
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§6-2 The Fundamental Theorem of Calculus
Applications of Definite Integrals
Example (Useful life)
An amusement company maintains records for each video game it installs in an arcade. Suppose that C(t) and R(t) represent the total accumulated costs and revenues (in thousands of dollars), respectively, t years after a particular game has been installed and that C (t) = 2 , R (t) = 9e−0.5t .
Example (Useful life)
An amusement company maintains records for each video game it installs in an arcade. Suppose that C(t) and R(t) represent the total accumulated costs and revenues (in thousands of dollars), respectively, t years after a particular game has been installed and that C (t) = 2 , R (t) = 9e−0.5t .
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The value of t for which C (t) = R (t) is called the useful life of the game. (A) Find the useful life of the game to the nearest year. (B) Find the total profit accumulated during the useful life of the game.
The company is currently manufacturing 1, 500 sets per month, but is planning to increase production. Find the change in the monthly profit if monthly production is increased to 1, 600 sets.
§6-2 The Fundamental Theorem of Calculus
§6-2 The Fundamental Theorem of Calculus
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§6-2 The Fundamental Theorem of Calculus
The value of t for which C (t) = R (t) is called the useful life of the game.
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§6-2 The Fundamental Theorem of Calculus
Applications of Definite Integrals
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§6-2 The Fundamental Theorem of Calculus
Applications of Definite Integrals
Solution: (A). R (t) = C (t) (B)
3
=⇒
9e−0.5t = 2
=⇒
t≈3
P(3) − P(0) =
where R(t) is the rate of production (in thousands of barrels per year) t years after pumping begins. Approximately how many barrels of oil will the field produce during the first 10 years of production?
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§6-2 The Fundamental Theorem of Calculus
Applications of Definite Integrals
Solution: We want to know P(1600) − P(1500):
1600
P(1600) − P(1500) =
References
Thomas’ Calculus: §5-4.
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§6-2 The Fundamental Theorem of Calclege Mathematics for Business: §12-5: 11. 27. 31. 35. 61. 63. 67.
Applications of Definite Integrals
Example (Oil Production)
Using data from the first 3 years of production as well as geological studies, the management of an oil company estimates that oil will be pumped from a producing field at a rate given by R(t) = 100 + 5, t+1 0 ≤ t ≤ 20 ,
The value of t for which C (t) = R (t) is called the useful life of the game. (A) Find the useful life of the game to the nearest year.
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§6-2 The Fundamental Theorem of Calculus
Applications of Definite Integrals
Solution: Knowing the rate of pumping, the amount of pumping during the first 10 years is
0 3
P (t)dt (R (t) − C (t))dt
0
=
= 12 − 18e−1.5 ≈ $7, 984 . (See College Mathematics for Business, §12-5, Example 6.)
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§6-2 The Fundamental Theorem of Calculus
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§6-2 The Fundamental Theorem of Calculus
Applications of Definite Integrals
Example (Change in Profit)
A company manufactures x TV sets per month. The monthly marginal profit (in dollars) is given by P (x) = 165 − 0.1x , 0 ≤ x ≤ 4, 000 .
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§6-2 The Fundamental Theorem of Calculus
Applications of Definite Integrals
Example (Useful life)
An amusement company maintains records for each video game it installs in an arcade. Suppose that C(t) and R(t) represent the total accumulated costs and revenues (in thousands of dollars), respectively, t years after a particular game has been installed and that C (t) = 2 , R (t) = 9e−0.5t .
10 0
R(t)dt =
100 + 5 dt t+1 0 = [100 ln(t + 1) + 5t]10 0 = 100 ln 11 + 50 ≈ 289.7895 .
10
Thus, the field will produce about 289,790 barrels of oil during the first 10 years.